Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
99 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10624833 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C243⋊C3 | C3 | 1 | 8 | yes | maximal, complemented |
| 3 | C3×C243 | C3 | 1 | 4 | yes | maximal, complemented |
| 4 | #6903 | C9 | 1 | 1 | yes | characteristic |
| 5 | C81⋊C3 | C9 | 1 | 8 | yes | — |
| 6 | C3×C81 | C32 | 1 | 4 | yes | — |
| 7 | C243 | C32 | 1 | 8 | yes | — |
| 8 | C243 | C32 | 1 | 1 | yes | characteristic |
| 9 | #737 | C27 | 1 | 1 | yes | characteristic |
| 10 | C27⋊C3 | C27 | 1 | 8 | yes | — |
| 11 | C3×C27 | C3×C9 | 1 | 4 | yes | — |
| 12 | C81 | C3×C9 | 1 | 8 | yes | — |
| 13 | C81 | C33 | 1 | 1 | yes | characteristic |
| 14 | 3−1+2 | C81 | 1 | 8 | yes | — |
| 15 | H(3,𝔽3) | C81 | 1 | 1 | yes | characteristic |
| 16 | C3×C9 | C3×C27 | 1 | 4 | yes | — |
| 17 | C27 | C3×C27 | 1 | 8 | yes | — |
| 18 | C27 | C32×C9 | 1 | 1 | yes | characteristic |
| 19 | C32 | C3×C81 | 1 | 4 | yes | — |
| 20 | C9 | C3×C81 | 1 | 8 | yes | — |
| 21 | C9 | C32×C27 | 1 | 1 | yes | characteristic |
| 22 | C3 | — | 3 | 4 | no | — |
| 23 | C3 | C32×C81 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C243 | C32 |
| commutator, socle | C3 | C32×C81 |
| Frattini, cosocle | C81 | C33 |
| prime radical | H(3,𝔽3) | C81 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C243 ◁ G | |
| LECS | G ▷ C81 ▷ C27 ▷ C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C243⋊C3 | C3 | semidirect |
| 3. C3×C243 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624833 | 3 |
| C243⋊C3 | 3 |
| C3×C243 | 3 |